Answer the following macro quizz

2 Risk Sharing Productivity: Consumption and Labor Consider a social planner who maximizes the utility of a continuum of agents of measure one ex-ante identical, with preferences over consumption and leisure u (c, !) where we assume that utility function a (.) is strictly concave, continuously differentiable and increasing in both its arguments. Agents are heterogeneous ex-post according to their productivity. They draw one non- negative productivity parameter @ 6 2 independently from a known continuous distribution function F. We assume that the support ? = ["min, "max] is compact and the density function density function f (w) is bounded above and bounded away from zero. The resource constraint in this economy is ( c(w) f ( w)dwa / (1 - 1( w) )wf ( w ) dw te for some endowment e 2 0. Let us assume that agents' productivity is ex-post publicly observable, so that consump- tion and leisure can be conditioned on their ability, i.e. c(w) and I (w). (a) Write down the social planner problem, noting that, given that the agents are a continuum of measure one, by the law of large number, the ex-ante probability coincides with the ex-post distribution. (b) Show that, under the given assumptions, leisure is a decreasing function of the pro- ductivity parameter w. Discuss. (c) Show that the dependence of consumption on the productivity parameters depends on the assumption on the sign of the cross-derivative un (short notation for 2 ). Comment. (d) Show that assuming that leisure is a normal good is sufficient (for any sign of ua) to have the utility function decreasing in the productivity. Discuss the role of risk sharing. 2 (e) Assume now that agents' productivity is not ex-post observable anymore. Given your result in the previous point, is the previous allocation implementable? Comment.3 Overlapping Generations In this problem we study an overlapping generations model under two market interpretations and isolate the conditions for Pareto optimality. All individuals live for only two periods. Generation t (denoted by the superscript) has utility function: u (c) + Bu (C+1) where u : R4 - R is strictly concave and increasing in the single consumption good c. Individuals are endowed with one unit of labor in the first period of their lives and supply it inelastically. Capital is owned by individuals and rented out to firms. Competitive firms rent capital and labor at prices r, and w (in terms of time t consumption goods). At time zero all initial capital is held by the old (i.e. generation t = -1). The resource constraint is, hittate F ( ke, 1 ) + ( 1 -8) ke, where F is a constant returns to scale (CRS) production function and 6 e (0, 1]. (a) Sequential Trade. Consider the sequential competitive market arrangement where individuals in generation t face the budget constraints, with Re = (1 - 6 + r;). Given ko, define a competitive equilibrium for this market arrangement for , { (C, 41) , kat1}. and prices {rt, we}=0- (b) Time-0 trade. Now consider the complete market arrangement where we imagine all generations (the born and yet unborn) and firms meeting at time zero and competitively trading in claims for future consumption, labor and capital. We generalize the notation and specialize it to interpret our overlapping generations model.Generation t faces the budget constraint: $=0 $=0 where we have normalize qg = 1 and R, is as before. Notation: n, and ko represent endowment of labor in period s and initial capital owned by generation t. Thus, in our OLG model: n; = 1 for s = t and n; = 0 for s # t; kol = ko and ky = 0 for s # -1. Think of each generation-t as having a utility function Ut defined over the entire con- sumption stream { }. Of course, in our OLG model: U' ({}2,) = u(c) + Bu (C+1). Define an equilibrium for 2, { (d, c) , kit}, and prices (of ), and {re, we)in using the standard Walrasian setup. Show that equilibria must satisfy the arbitrage condition: q/q+, = R4+1. Argue that the sequential market equilibrium in part (a) is a time-0 market equilibrium, as outlined here, for appropriately chosen prices {q/}- (c) Consider the special case of log utility, u (c) = logic, and Cobb-Douglas production function F (nt, ke) = Aken;-". Characterize the entire equilibrium allocation , { (c, ci+1) , ke+1 ) . and prices {rt, we}- Solve for the steady state level of capital kys. Show that the equilibrium is not Pareto efficient if steady state capital higher than the golden rule k, = arg maxx {F(k, 1) - ok}. Show that there are parameters values for which kg k, for s > t). (d) In terms of the time-0, complete-market arrangement. Why does the First Welfare Theorem fail to apply? (Hint: argue that the condition $=0 is necessary for the proof of the first welfare theorem and that it is not satisfied here). Show that the welfare theorem does apply if kiss Sky. (e) The previous points show that if kas 0. Use this and the results above to show that when kes